Tetragonal disphenoid tetrahedral honeycomb | |
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Type | convex uniform honeycomb dual |
Coxeter-Dynkin diagram | |
Cell type | Tetragonal disphenoid |
Face types | isosceles triangle {3} |
Vertex figure | tetrakis hexahedron |
Space group | Im3m (229) |
Symmetry | [[4, 3, 4]] |
Coxeter group | , [4, 3, 4] |
Dual | Bitruncated cubic honeycomb |
Properties | cell-transitive, face-transitive, vertex-transitive |
The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an oblate tetrahedrille or shortened to obtetrahedrille. [1]
A cell can be seen as 1/12 of a translational cube, with its vertices centered on two faces and two edges. Four of its edges belong to 6 cells, and two edges belong to 4 cells.
The tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb.
Its vertices form the A*
3 / D*
3 lattice, which is also known as the body-centered cubic lattice.
This honeycomb's vertex figure is a tetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a rhombic dodecahedron. Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, and is surrounded by four disphenoids, they form an irregular octahedron. When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of parallelepiped called a trigonal trapezohedron.
An orientation of the tetragonal disphenoid honeycomb can be obtained by starting with a cubic honeycomb, subdividing it at the planes , , and (i.e. subdividing each cube into path-tetrahedra), then squashing it along the main diagonal until the distance between the points (0, 0, 0) and (1, 1, 1) becomes the same as the distance between the points (0, 0, 0) and (0, 0, 1).
Hexakis cubic honeycomb Pyramidille [2] | |
---|---|
Type | Dual uniform honeycomb |
Coxeter–Dynkin diagrams | |
Cell | Isosceles square pyramid |
Faces | Triangle square |
Space group Fibrifold notation | Pm3m (221) 4−:2 |
Coxeter group | , [4, 3, 4] |
vertex figures | , |
Dual | Truncated cubic honeycomb |
Properties | Cell-transitive |
The hexakis cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it a pyramidille. [3]
Cells can be seen in a translational cube, using 4 vertices on one face, and the cube center. Edges are colored by how many cells are around each of them.
It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells.
There are two types of planes of faces: one as a square tiling, and flattened triangular tiling with half of the triangles removed as holes.
Tiling plane | ||
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Symmetry | p4m, [4,4] (*442) | pmm, [∞,2,∞] (*2222) |
It is dual to the truncated cubic honeycomb with octahedral and truncated cubic cells:
If the square pyramids of the pyramidille are joined on their bases, another honeycomb is created with identical vertices and edges, called a square bipyramidal honeycomb, or the dual of the rectified cubic honeycomb.
It is analogous to the 2-dimensional tetrakis square tiling:
Square bipyramidal honeycomb Oblate octahedrille [4] | |
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Type | Dual uniform honeycomb |
Coxeter–Dynkin diagrams | |
Cell | Square bipyramid |
Faces | Triangles |
Space group Fibrifold notation | Pm3m (221) 4−:2 |
Coxeter group | , [4,3,4] |
vertex figures | , |
Dual | Rectified cubic honeycomb |
Properties | Cell-transitive, Face-transitive |
The square bipyramidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it an oblate octahedrille or shortened to oboctahedrille. [5]
A cell can be seen positioned within a translational cube, with 4 vertices mid-edge and 2 vertices in opposite faces. Edges are colored and labeled by the number of cells around the edge.
It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells. The original cubic honeycomb walls are removed, joining pairs of square pyramids into square bipyramids (octahedra). Its vertex and edge framework is identical to the hexakis cubic honeycomb.
There is one type of plane with faces: a flattened triangular tiling with half of the triangles as holes. These cut face-diagonally through the original cubes. There are also square tiling plane that exist as nonface holes passing through the centers of the octahedral cells.
Tiling plane | Square tiling "holes" | flattened triangular tiling |
---|---|---|
Symmetry | p4m, [4,4] (*442) | pmm, [∞,2,∞] (*2222) |
It is dual to the rectified cubic honeycomb with octahedral and cuboctahedral cells:
This section may be confusing or unclear to readers. In particular, How is this different from subdividing a cube into only six tetrahedra and then translating? And what is the justification for describing it in an article about a different honeycomb?. (May 2018) (Learn how and when to remove this template message) |
Phyllic disphenoidal honeycomb Eighth pyramidille [6] | |
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(No image) | |
Type | Dual uniform honeycomb |
Coxeter-Dynkin diagrams | |
Cell | Phyllic disphenoid |
Faces | Rhombus Triangle |
Space group Fibrifold notation Coxeter notation | Im3m (229) 8o:2 [[4,3,4]] |
Coxeter group | [4,3,4], |
vertex figures | , |
Dual | Omnitruncated cubic honeycomb |
Properties | Cell-transitive, face-transitive |
The phyllic disphenoidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls this an Eighth pyramidille. [7]
A cell can be seen as 1/48 of a translational cube with vertices positioned: one corner, one edge center, one face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.
It is dual to the omnitruncated cubic honeycomb:
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.
In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6}.
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.
In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – consists of four such units of the cubic honeycomb.
The order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.
In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.
The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.
In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.
In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.
In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In geometry, the ten-of-diamonds decahedron is a space-filling polyhedron with 10 faces, 2 opposite rhombi with orthogonal major axes, connected by 8 identical isosceles triangle faces. Although it is convex, it is not a Johnson solid because its faces are not composed entirely of regular polygons. Michael Goldberg named it after a playing card, as a 10-faced polyhedron with two opposite rhombic (diamond-shaped) faces. He catalogued it in a 1982 paper as 10-II, the second in a list of 26 known space-filling decahedra.